{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "reverse-interview",
   "metadata": {},
   "source": [
    "# Differentiation in Python: Symbolic, Numerical and Automatic"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "parental-conclusion",
   "metadata": {},
   "source": [
    "In this lab you explore which tools and libraries are available in Python to compute derivatives. You will perform symbolic differentiation with `SymPy` library, numerical with `NumPy` and automatic with `JAX` (based on `Autograd`). Comparing the speed of calculations, you will investigate the computational efficiency of those three methods."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "looking-barcelona",
   "metadata": {},
   "source": [
    "# Table of Contents\n",
    "- [ 1 - Functions in Python](#1)\n",
    "- [ 2 - Symbolic Differentiation](#2)\n",
    "  - [ 2.1 - Introduction to Symbolic Computation with `SymPy`](#2.1)\n",
    "  - [ 2.2 - Symbolic Differentiation with `SymPy`](#2.2)\n",
    "  - [ 2.3 - Limitations of Symbolic Differentiation](#2.3)\n",
    "- [ 3 - Numerical Differentiation](#3)\n",
    "  - [ 3.1 - Numerical Differentiation with `NumPy`](#3.1)\n",
    "  - [ 3.2 - Limitations of Numerical Differentiation](#3.2)\n",
    "- [ 4 - Automatic Differentiation](#4)\n",
    "  - [ 4.1 - Introduction to `JAX`](#4.1)\n",
    "  - [ 4.2 - Automatic Differentiation with `JAX` ](#4.2)\n",
    "- [ 5 - Computational Efficiency of Symbolic, Numerical and Automatic Differentiation](#5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "101116ab",
   "metadata": {},
   "source": [
    "<a name='1'></a>\n",
    "## 1 - Functions in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc6140d8",
   "metadata": {},
   "source": [
    "This is just a reminder how to define functions in Python. A simple function $f\\left(x\\right) = x^2$, it can be set up as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d07a15ef",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9\n"
     ]
    }
   ],
   "source": [
    "def f(x):\n",
    "    return x**2\n",
    "\n",
    "print(f(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "06330bd1",
   "metadata": {},
   "source": [
    "You can easily find the derivative of this function analytically. You can set it up as a separate function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1ff4ffb5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6\n"
     ]
    }
   ],
   "source": [
    "def dfdx(x):\n",
    "    return 2*x\n",
    "\n",
    "print(dfdx(3))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8301af3f",
   "metadata": {},
   "source": [
    "Since you have been working with the `NumPy` arrays, you can apply the function to each element of an array:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9f5831d8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x: \n",
      " [1 2 3]\n",
      "f(x) = x**2: \n",
      " [1 4 9]\n",
      "f'(x) = 2x: \n",
      " [2 4 6]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "x_array = np.array([1, 2, 3])\n",
    "\n",
    "print(\"x: \\n\", x_array)\n",
    "print(\"f(x) = x**2: \\n\", f(x_array))\n",
    "print(\"f'(x) = 2x: \\n\", dfdx(x_array))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8428f910",
   "metadata": {},
   "source": [
    "Now you can apply those functions `f` and `dfdx` to an array of a larger size. The following code will plot function and its derivative (you don't have to understand the details of the `plot_f1_and_f2` function at this stage):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "5c255f4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Output of plotting commands is displayed inline within the Jupyter notebook.\n",
    "%matplotlib inline\n",
    "\n",
    "def plot_f1_and_f2(f1, f2=None, x_min=-5, x_max=5, label1=\"f(x)\", label2=\"f'(x)\"):\n",
    "    x = np.linspace(x_min, x_max,100)\n",
    "\n",
    "    # Setting the axes at the centre.\n",
    "    fig = plt.figure()\n",
    "    ax = fig.add_subplot(1, 1, 1)\n",
    "    ax.spines['left'].set_position('center')\n",
    "    ax.spines['bottom'].set_position('zero')\n",
    "    ax.spines['right'].set_color('none')\n",
    "    ax.spines['top'].set_color('none')\n",
    "    ax.xaxis.set_ticks_position('bottom')\n",
    "    ax.yaxis.set_ticks_position('left')\n",
    "\n",
    "    plt.plot(x, f1(x), 'r', label=label1)\n",
    "    if not f2 is None:\n",
    "        # If f2 is an array, it is passed as it is to be plotted as unlinked points.\n",
    "        # If f2 is a function, f2(x) needs to be passed to plot it.        \n",
    "        if isinstance(f2, np.ndarray):\n",
    "            plt.plot(x, f2, 'bo', markersize=3, label=label2,)\n",
    "        else:\n",
    "            plt.plot(x, f2(x), 'b', label=label2)\n",
    "    plt.legend()\n",
    "\n",
    "    plt.show()\n",
    "    \n",
    "plot_f1_and_f2(f, dfdx)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffb711b8",
   "metadata": {},
   "source": [
    "In real life the functions are more complicated and it is not possible to calculate the derivatives analytically every time. Let's explore which tools and libraries are available in Python for the computation of derivatives without manual derivation."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d17f76f",
   "metadata": {},
   "source": [
    "<a name='2'></a>\n",
    "## 2 - Symbolic Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "severe-studio",
   "metadata": {},
   "source": [
    "**Symbolic computation** deals with the computation of mathematical objects that are represented exactly, not approximately (e.g. $\\sqrt{2}$ will be written as it is, not as $1.41421356237$). For differentiation it would mean that the output will be somehow similar to if you were computing derivatives by hand using rules (analytically). Thus, symbolic differentiation can produce exact derivatives."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b51ca07",
   "metadata": {},
   "source": [
    "<a name='2.1'></a>\n",
    "### 2.1 - Introduction to Symbolic Computation with `SymPy`\n",
    "\n",
    "Let's explore symbolic differentiation in Python with commonly used `SymPy` library.\n",
    "\n",
    "If you want to compute the approximate decimal value of $\\sqrt{18}$, you could normally do it in the following way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "52d0b0a6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.242640687119285"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import math\n",
    "\n",
    "math.sqrt(18)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a5227c24",
   "metadata": {},
   "source": [
    "The output $4.242640687119285$ is an approximate result. You may recall that $\\sqrt{18} = \\sqrt{9 \\cdot 2} = 3\\sqrt{2}$ and see that it is pretty much impossible to deduct it from the approximate result. But with the symbolic computation systems the roots are not approximated with a decimal number but rather only simplified, so the output is exact:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "d5f1747d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 \\sqrt{2}$"
      ],
      "text/plain": [
       "3*sqrt(2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# This format of module import allows to use the sympy functions without sympy. prefix.\n",
    "from sympy import *\n",
    "\n",
    "# This is actually sympy.sqrt function, but sympy. prefix is omitted.\n",
    "sqrt(18)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5495348",
   "metadata": {},
   "source": [
    "Numerical evaluation of the result is available, and you can set number of the digits to show in the approximated output:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "925375b9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4.2426407$"
      ],
      "text/plain": [
       "4.2426407"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(sqrt(18),8)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb625f44",
   "metadata": {},
   "source": [
    "In `SymPy` variables are defined using **symbols**. In this particular library they need to be predefined (a list of them should be provided). Have a look in the cell below, how the symbolic expression, correspoinding to the mathematical expression $2x^2 - xy$, is defined:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "b52721c0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x^{2} - x y$"
      ],
      "text/plain": [
       "2*x**2 - x*y"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# List of symbols.\n",
    "x, y = symbols('x y')\n",
    "# Definition of the expression.\n",
    "expr = 2 * x**2 - x * y\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "edffde06",
   "metadata": {},
   "source": [
    "Now you can perform various manipulations with this expression: add or subtract some terms, multiply by other expressions etc., just like if you were doing it by hands:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "ac575c24",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(x^{3} + 2 x^{2}\\right)$"
      ],
      "text/plain": [
       "x*(x**3 + 2*x**2)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_manip = x * (expr + x * y + x**3)\n",
    "expr_manip"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db151265",
   "metadata": {},
   "source": [
    "You can also expand the expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "afb04770",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{4} + 2 x^{3}$"
      ],
      "text/plain": [
       "x**4 + 2*x**3"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(expr_manip)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77b3750c",
   "metadata": {},
   "source": [
    "Or factorise it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "0fc456cb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} \\left(x + 2\\right)$"
      ],
      "text/plain": [
       "x**3*(x + 2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(expr_manip)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c885c385",
   "metadata": {},
   "source": [
    "To substitute particular values for the variables in the expression, you can use the following code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "3c7d239e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4.0$"
      ],
      "text/plain": [
       "4.00000000000000"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.evalf(subs={x:-1, y:2})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4acf4536",
   "metadata": {},
   "source": [
    "This can be used to evaluate a function $f\\left(x\\right) = x^2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "622e4335",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9.0$"
      ],
      "text/plain": [
       "9.00000000000000"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_symb = x ** 2\n",
    "f_symb.evalf(subs={x:3})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1346fb9f",
   "metadata": {},
   "source": [
    "You might be wondering now, is it possible to evaluate the symbolic functions for each element of the array? At the beginning of the lab you have defined a `NumPy` array `x_array`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "af6562d3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 2 3]\n"
     ]
    }
   ],
   "source": [
    "print(x_array)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f13ae4e0",
   "metadata": {},
   "source": [
    "Now try to evaluate function `f_symb` for each element of the array. You will get an error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "11df38ae",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "'Pow' object is not callable\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    f_symb(x_array)\n",
    "except TypeError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f7800fe",
   "metadata": {},
   "source": [
    "It is possible to evaluate the symbolic functions for each element of the array, but you need to make a function `NumPy`-friendly first:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "3d7d8a58",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.utilities.lambdify import lambdify\n",
    "\n",
    "f_symb_numpy = lambdify(x, f_symb, 'numpy')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "631e3e71",
   "metadata": {},
   "source": [
    "The following code should work now:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "1ab8f0da",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x: \n",
      " [1 2 3]\n",
      "f(x) = x**2: \n",
      " [1 4 9]\n"
     ]
    }
   ],
   "source": [
    "print(\"x: \\n\", x_array)\n",
    "print(\"f(x) = x**2: \\n\", f_symb_numpy(x_array))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "681062e1",
   "metadata": {},
   "source": [
    "`SymPy` has lots of great functions to manipulate expressions and perform various operations from calculus. More information about them can be found in the official documentation [here](https://docs.sympy.org/)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c6c9d29",
   "metadata": {},
   "source": [
    "<a name='2.2'></a>\n",
    "### 2.2 - Symbolic Differentiation with `SymPy`\n",
    "\n",
    "Let's try to find a derivative of a simple power function using `SymPy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "abb93aeb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 3 x^{2}$"
      ],
      "text/plain": [
       "3*x**2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**3,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd8a81b1",
   "metadata": {},
   "source": [
    "Some standard functions can be used in the expression, and `SymPy` will apply required rules (sum, product, chain) to calculate the derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "63ce2cd2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9 \\cos{\\left(3 x \\right)} - 2 e^{- 2 x}$"
      ],
      "text/plain": [
       "9*cos(3*x) - 2*exp(-2*x)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfdx_composed = diff(exp(-2*x) + 3*sin(3*x), x)\n",
    "dfdx_composed"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "13407c2b",
   "metadata": {},
   "source": [
    "Now calculate the derivative of the function `f_symb` defined in [2.1](#2.1) and make it `NumPy`-friendly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "597d5879",
   "metadata": {},
   "outputs": [],
   "source": [
    "dfdx_symb = diff(f_symb, x)\n",
    "dfdx_symb_numpy = lambdify(x, dfdx_symb, 'numpy')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f9bee28",
   "metadata": {},
   "source": [
    "Evaluate function `dfdx_symb_numpy` for each element of the `x_array`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "b74b4d04",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x: \n",
      " [1 2 3]\n",
      "f'(x) = 2x: \n",
      " [2 4 6]\n"
     ]
    }
   ],
   "source": [
    "print(\"x: \\n\", x_array)\n",
    "print(\"f'(x) = 2x: \\n\", dfdx_symb_numpy(x_array))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ada41a99",
   "metadata": {},
   "source": [
    "You can apply symbolically defined functions to the arrays of larger size. The following code will plot function and its derivative, you can see that it works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "031a757c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_f1_and_f2(f_symb_numpy, dfdx_symb_numpy)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de01feee",
   "metadata": {},
   "source": [
    "<a name='2.3'></a>\n",
    "### 2.3 - Limitations of Symbolic Differentiation\n",
    "\n",
    "Symbolic Differentiation seems to be a great tool. But it also has some limitations. Sometimes the output expressions are too complicated and even not possible to evaluate. For example, find the derivative of the function $$\\left|x\\right| = \\begin{cases} x, \\ \\text{if}\\ x > 0\\\\  -x, \\ \\text{if}\\ x < 0 \\\\ 0, \\ \\text{if}\\ x = 0\\end{cases}$$ Analytically, its derivative is:\n",
    "$$\\frac{d}{dx}\\left(\\left|x\\right|\\right) = \\begin{cases} 1, \\ \\text{if}\\ x > 0\\\\  -1, \\ \\text{if}\\ x < 0\\\\\\ \\text{does not exist}, \\ \\text{if}\\ x = 0\\end{cases}$$\n",
    "\n",
    "Have a look the output from the symbolic differentiation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "collect-needle",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\operatorname{re}{\\left(x\\right)} \\frac{d}{d x} \\operatorname{re}{\\left(x\\right)} + \\operatorname{im}{\\left(x\\right)} \\frac{d}{d x} \\operatorname{im}{\\left(x\\right)}\\right) \\operatorname{sign}{\\left(x \\right)}}{x}$"
      ],
      "text/plain": [
       "(re(x)*Derivative(re(x), x) + im(x)*Derivative(im(x), x))*sign(x)/x"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfdx_abs = diff(abs(x),x)\n",
    "dfdx_abs"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9c3d8e5",
   "metadata": {},
   "source": [
    "Looks complicated, but it would not be a problem if it was possible to evaluate. But check, that for $x=-2$ instead of the derivative value $-1$ it outputs some unevaluated expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "d53e7c64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\operatorname{re}{\\left(x\\right)} \\frac{d}{d x} \\operatorname{re}{\\left(x\\right)} + \\operatorname{im}{\\left(x\\right)} \\frac{d}{d x} \\operatorname{im}{\\left(x\\right)}\\right) \\operatorname{sign}{\\left(x \\right)}}{x}$"
      ],
      "text/plain": [
       "(re(x)*Derivative(re(x), x) + im(x)*Derivative(im(x), x))*sign(x)/x"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dfdx_abs.evalf(subs={x:-2})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4a9140c",
   "metadata": {},
   "source": [
    "And in the `NumPy` friendly version it also will give an error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "644f00ce",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "name 'Derivative' is not defined\n"
     ]
    }
   ],
   "source": [
    "dfdx_abs_numpy = lambdify(x, dfdx_abs,'numpy')\n",
    "\n",
    "try:\n",
    "    dfdx_abs_numpy(np.array([1, -2, 0]))\n",
    "except NameError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e1f3c94",
   "metadata": {},
   "source": [
    "In fact, there are problems with the evaluation of the symbolic expressions wherever there is a \"jump\" in the derivative (e.g. function expressions are different for different intervals of $x$), like it happens with $\\frac{d}{dx}\\left(\\left|x\\right|\\right)$. \n",
    "\n",
    "Also, you can see in this example, that you can get a very complicated function as an output of symbolic computation. This is called **expression swell**, which results in unefficiently slow computations. You will see the example of that below after learning other differentiation libraries in Python."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1cb85963",
   "metadata": {},
   "source": [
    "<a name='3'></a>\n",
    "## 3 - Numerical Differentiation\n",
    "\n",
    "This method does not take into account the function expression. The only important thing is that the function can be evaluated in the nearby points $x$ and $x+\\Delta x$, where $\\Delta x$ is sufficiently small. Then $\\frac{df}{dx}\\approx\\frac{f\\left(x + \\Delta x\\right) - f\\left(x\\right)}{\\Delta x}$, which can be called a **numerical approximation** of the derivative. \n",
    "\n",
    "Based on that idea there are different approaches for the numerical approximations, which somehow vary in the computation speed and accuracy. However, for all of the methods the results are not accurate - there is a round off error. At this stage there is no need to go into details of various methods, it is enough to investigate one of the numerial differentiation functions, available in `NumPy` package."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1cc2a87e",
   "metadata": {},
   "source": [
    "<a name='3.1'></a>\n",
    "### 3.1 - Numerical Differentiation with `NumPy`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c469b76c",
   "metadata": {},
   "source": [
    "You can call function `np.gradient` to find the derivative of function $f\\left(x\\right) = x^2$ defined above. The first argument is an array of function values, the second defines the spacing $\\Delta x$ for the evaluation. Here pass it as an array of $x$ values, the differences will be calculated automatically. You can find the documentation [here](https://numpy.org/doc/stable/reference/generated/numpy.gradient.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "b275519f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_array_2 = np.linspace(-5, 5, 100)\n",
    "dfdx_numerical = np.gradient(f(x_array_2), x_array_2)\n",
    "\n",
    "plot_f1_and_f2(dfdx_symb_numpy, dfdx_numerical, label1=\"f'(x) exact\", label2=\"f'(x) approximate\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a1d5843",
   "metadata": {},
   "source": [
    "Try to do numerical differentiation for more complicated function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "9fa0d7cf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BjSUQcMVBIiJyRpwzUJ1Ll4DgYODiRRg/242cJlFccZCcEucMEBFPLazOl18CFy8C4eEwPNoHBs2tX0JEROSIeJigOqtWWe5HjwY0TAJEROS8GAaqcvIksG2b5etRo9TthYiIqJ4xDFTlgw8AEWDAACA0VO1uiIiI6hXDwM1EgPfft3wdF6dqK0RERA2BEwhvYly3D9nZrRDpDRgee0ztdoiIiOodRwYqSEoCQh69y3IdgquZSFrrp3ZLRERE9Y7rDNxgNAIhIQKz+bczB3gdAnIFXGeAiDgycEN2NqyCAMDrEBARkWtgGLghMhLQasxW23gdAiIicgUMAzcYWgmW+z8PHUoB8DoERETkOjhnoNy+fUDv3jB6RyLnq8OI6OTBIEAugXMGiIinFpZbvx4AYHioKwyDPVRuhoiIqOHwMEG5G2EAw4ap2wcREVEDYxgALNci2L8f0GqBhx9Wuxuyc6+88gr69u0LHx8fBAQEVFlz8uRJDB06FD4+PmjevDmef/55lJaWWtVs374dd911Fzw9PREREYFV5RfHqmDp0qUIDQ2Fl5cXoqKisGfPHqvnr169iokTJ6JJkybw8/PDY489hjNnztjqrRKRi2AYAIANGyz3ffsCzZqp2wvZvWvXrmHEiBH461//WuXzZWVlGDp0KK5du4Zdu3bh/fffx6pVq/Diiy8qNcePH8fQoUPxwAMPYP/+/Zg2bRr+8pe/YMuWLUrN2rVrMX36dCQkJODHH39E9+7dERMTg4KCAqXm2WefxYYNG/Dpp5/i22+/xS+//IJHH320/t48ETknIZHBg0UAkYUL1e6EHMjKlStFr9dX2r5p0ybRarWSn5+vbFu2bJn4+/tLSUmJiIjMnDlTOnfubPW6J554QmJiYpTHffr0kYkTJyqPy8rKpGXLljJ//nwRESksLBR3d3f59NNPlZqjR48KAElNTa31+zCZTAJATCZTrV9DRM6FIwNFRUBKiuVrzhcgG0hNTUXXrl0RFBSkbIuJiUFRURGOHDmi1AwaNMjqdTExMUhNTQVgGX3Yt2+fVY1Wq8WgQYOUmn379uH69etWNR06dECbNm2UmqqUlJSgqKjI6kZEro1hYMsW4Pp1oF07oH17tbshJ5Cfn28VBAAoj/Pz82usKSoqwpUrV3D27FmUlZVVWVNxHx4eHpXmLVSsqcr8+fOh1+uVW+vWrW/rfRKR82AY4FkEBGD27NnQaDQ13jIzM9Vu0ybmzJkDk8mk3E6dOqV2S0SkMpdeZ8CYW4rsr0yIRCsYGAZc2owZMxAXF1djTXh4eK32FRwcXGnWf/kM/+DgYOX+5ln/Z86cgb+/P7y9vaHT6aDT6aqsqbiPa9euobCw0Gp0oGJNVTw9PeHp6Vmr90JErsFlRwaSkoCQtjoMvLgeITiBpKN91W6JVNSsWTN06NChxpuHR+0Wo4qOjsahQ4esZv0nJyfD398fnTp1Umq2bt1q9brk5GRER0cDADw8PNCrVy+rGrPZjK1btyo1vXr1gru7u1VNVlYWTp48qdQQEdWK2jMY1XDqlIhWazmBoPym01m2E93KiRMnJD09XV566SXx8/OT9PR0SU9Pl+LiYhERKS0tlS5dusjgwYNl//79snnzZmnWrJnMmTNH2cfPP/8sPj4+8vzzz8vRo0dl6dKlotPpZPPmzUrNmjVrxNPTU1atWiUZGRkybtw4CQgIsDpLYcKECdKmTRvZtm2b/PDDDxIdHS3R0dF1ej88m4CIXDIMbNtmHQTKbykpandGjmD06NECoNItpcIPUG5urjz00EPi7e0tTZs2lRkzZsj169et9pOSkiI9evQQDw8PCQ8Pl5UrV1b6XkuWLJE2bdqIh4eH9OnTR3bv3m31/JUrV+SZZ56Rxo0bi4+Pj/zxj3+U06dP1+n9MAwQkUteqMhoBEJCBGazRtmm0wG5ubxKIbkeXqiIiFxyzoDBACx/5gAvV0xERAQXDQMAEO/+H+QiFCm/fx25uUB8vNodERERqcN1Ty3cvh0G5MHwVEuAIwJEROTCXHNkoLDQcpVCAOjfX81OiIiIVOeaYeC77ywnELRrB7RsqXY3REREqnLNMLB9u+WeowJEREQuHgYGDFCzCyIiIrvgemGA8wWIiIisuF4Y+P57wGwGIiOBVq3U7oaIiEh1rhcGOF+AiIjIiuuGAc4XICIiAuBqYcBkAtLTLV9zZICIiAiAq4WB8vkCERG8EAEREdENrhUGOF+AiIioEpcKA8bko0jBABi7Pax2K0RERHbDZS5UlPT/rmDcgXUwQwfts4LlvrxSIREREQBoRETUbqK+GY1ASIjAbNYo23Q6IDeXUweIioqKoNfrYTKZ4O/vr3Y7RKQClzhMkJ0NqyAAAGVlQE6OSg0RERHZEZcIA5GRgBZmq206neWkAiIiIlfnEmHAYACWN5kDHUoBWIJAYiIPERAREQEuMmcAv/4KNG8OI1ohZ0MmInr4MQgQ3cA5A0TkGmcT7NsHADC084Xh934qN0NERGRfXOIwAX74wXLfu7e6fRAREdkhhgEiIiIXxzBARETk4pw/DJw+DeTlARoN0LOn2t0QERHZHecPAzcmD6JjR8CPkweJiIhu5vxhgIcIiIiIasQwQERE5OKcOwyIMAwQERHdgnOHgbw84MwZy/rD3bur3Q0REZFdcu4wUD4q0KUL4OOjbi9ERER2yjXCAA8REBERVYthgIiIyMU5bxgQAfbutXzNMEBERFQt5w0DubnA+fOAuzvQtava3RAREdktp72EsXHLEWRjACI7eMPg6al2O0RERHbLKUcGkpKAkGcexkCkIOTwRiQlqd0RERGR/dKIiKjdhC0ZjUBICGA2/7ZNp7McNTAYVGuLyG4VFRVBr9fDZDLB399f7XaISAVONzKQnW0dBACgrAzIyVGnHyIiInvndGEgMhLQaq0HO3Q6ICJCpYaIiIjsnNOFAYMBWD4zBzqUArAEgcREHiIgIiKqjtOFAQCID0tBLkKRcvdM5OYC8fFqd0RERGS/nPPUwowMGJAHw31lAEcEiIiIauSUIwM4csRy36mTun0QERE5AOcMAxkZlvvOndXtg4iIyAE4XxgoLAR++cXyNUcGiIiIbsn5wkD5IQKDAeACKkRERLfkfGGAhwiIiIjqxPnCACcPEhER1YnzhQGODBAREdWJ84UBjgwQERHViXOFAZ5JQEREVGfOFQbKDxEYDIBer24vREREDsI5wwBHBYiIiGrNucJA+XwBTh4kIiKqNecKAxwZICIiqjPnCgMcGSAiIqoz5wkDJhOQl2f5umNHdXshIiJyIM4TBsoPEbRqBQQEqNoKERGRI3G+MMBDBERERHXiPGGAKw8SERHdFucLAxwZICIiqhPnCQM8rZCIiOi2OEcYMJlgNApSMABGPUcGiIiI6sIpwkDSq2cRghMYiBSEdNMjKUntjshZ5ebmIj4+HmFhYfD29kbbtm2RkJCAa9euWdUdPHgQ9913H7y8vNC6dWssXLiw0r4+/fRTdOjQAV5eXujatSs2bdpk9byI4MUXX0SLFi3g7e2NQYMGITs726rm/PnzGDlyJPz9/REQEID4+HhcvHjR9m+ciJyaw4cBoxEYtyAMZugAAGYzMH68ZTuRrWVmZsJsNiMxMRFHjhzBm2++iXfffRd/+9vflJqioiIMHjwYISEh2LdvHxYtWoS5c+di+fLlSs2uXbsQGxuL+Ph4pKenY/jw4Rg+fDgOHz6s1CxcuBCLFy/Gu+++i7S0NPj6+iImJgZXr15VakaOHIkjR44gOTkZGzduxI4dOzBu3LiG+TCIyHmIg9u2TQSofEtJUbszchULFy6UsLAw5fE777wjjRs3lpKSEmXbrFmzpH379srjxx9/XIYOHWq1n6ioKBk/fryIiJjNZgkODpZFixYpzxcWFoqnp6d8/PHHIiKSkZEhAGTv3r1KzTfffCMajUby8vJq3b/JZBIAYjKZav0aInIuDj8yEBkJaGG22qbTARERKjVELsdkMiEwMFB5nJqaivvvvx8eHh7KtpiYGGRlZeHChQtKzaBBg6z2ExMTg9TUVADA8ePHkZ+fb1Wj1+sRFRWl1KSmpiIgIAC9e/dWagYNGgStVou0tLRq+y0pKUFRUZHVjYhcm8OHAYMBWB6+ADqUArAEgcREy3ai+paTk4MlS5Zg/Pjxyrb8/HwEBQVZ1ZU/zs/Pr7Gm4vMVX1ddTfPmza2ed3NzQ2BgoFJTlfnz50Ov1yu31q1b1/r9EpFzcvgwAADxlxYjF6FISfwJublAfLzaHZGjmT17NjQaTY23zMxMq9fk5eVhyJAhGDFiBMaOHatS53U3Z84cmEwm5Xbq1Cm1WyIilbmp3cAdu3gROHMGBgCGJ4IAvdoNkSOaMWMG4uLiaqwJDw9Xvv7ll1/wwAMPoG/fvlYTAwEgODgYZ86csdpW/jg4OLjGmorPl29r0aKFVU2PHj2UmoKCAqt9lJaW4vz588rrq+Lp6QlPT88a3ysRuRbHHxn4+WfLfWAgoGcSoNvTrFkzdOjQocZb+RyAvLw8DBgwAL169cLKlSuh1Vr/bxQdHY0dO3bg+vXryrbk5GS0b98ejRs3Vmq2bt1q9brk5GRER0cDAMLCwhAcHGxVU1RUhLS0NKUmOjoahYWF2Ldvn1Kzbds2mM1mREVF2fDTISKnp/YMxjv2xReW0wfuvlvtTsgFGI1GiYiIkAcffFCMRqOcPn1auZUrLCyUoKAgefrpp+Xw4cOyZs0a8fHxkcTERKVm586d4ubmJq+99pocPXpUEhISxN3dXQ4dOqTULFiwQAICAmTdunVy8OBBeeSRRyQsLEyuXLmi1AwZMkR69uwpaWlp8v3330tkZKTExsbW6T3xbAIicvwwsGiRJQw8+aTanZALWLlypQCo8lbRgQMHpF+/fuLp6SmtWrWSBQsWVNrXJ598Iu3atRMPDw/p3LmzfP3111bPm81meeGFFyQoKEg8PT3lwQcflKysLKuac+fOSWxsrPj5+Ym/v7+MGTNGiouL6/SeGAaISCMiot64hA088wywbBnw978D//yn2t0QOZyioiLo9XqYTCb4+/ur3Q4RqcDx5wwcO2a5b9tW3T6IiIgclPOEgQozvYmIiKj2HDsMlJYCJ05YvubIABER0W1x7DBw6pQlEHh6Ai1bqt0NERGRQ3LsMFDxEIHWsd8KERGRWhz7LyjnCxAREd0x5wgDnC9ARER02xw7DJQvRcwwQEREdNscOwxwZICIiOiOOW4YEOGcASIiIhtw3DBw9ixQXAxoNEBYmNrdEBEROSzHDQPlowKtWgFeXur2QkRE5MAcNwxw8iAREZFNOG4Y4HwBIiIim3D8MMCRASIiojvCMEBEROTiHDcMcM4AERGRTThmGLhyBfjlF8vXDANERER3xDHDQPmogF4PNG6sbi9EREQOzjHDQMX5AhqNur0QERE5OMcMA5wvQEREZDOOGQZ4JgEREZHNOGQYMB4uRAoGwNi4q9qtEBEROTw3tRuoq6QkYNz2VTBDB+0cwfImQHy82l0RERE5Lo2IiNpN1JbRCISECMzm3yYN6nRAbi5gMKjXF5EjKyoqgl6vh8lkgr+/v9rtEJEKHOowQXY2rIIAAJSVATk5KjVERETkBBwqDERGAlqt9UCGTgdERKjUEBERkRNwqDBgMADLJx+GDqUALEEgMZGHCIiIiO6EQ4UBAIjvlIpchCLlnjnIzeXkQSIiojvlcGcTwGiEAXkw9DQBHBEgIiK6Yw43MoBTpyz3PDZARERkE44XBoxGyz3DABERkU04bhho3VrdPoiIiJyEY4UBER4mICIisjHHCgMmE3DpkuVrhgEiIiKbcKwwUD4q0KQJ4O2tbi9EREROwrHCACcPEhER2ZxjhYHykQFOHiQiIrIZxwoDHBkgIiKyOccMAxwZICIishnHCgM8rZCIiMjmHCsM8DABERGRzTlOGKi44BAPExAREdmM44SBigsOtWqlbi9EREROxHHCQMUFh3x81O2FiIjIiThOGOB8ASIionrBMEBEROTiHCcMcPIgERFRvXCcMMCRASIionrhOGGAIwNERET1wnHCAEcGiIiI6oVjhIGKCw4xDBAREdmUY4SBigsOMQwQERHZlGOEgfJDBIGBXHCIiIjIxhwjDHDyIBERUb1xjDDAyYNERET1xjHCACcPEhER1RvHCAPlIwM8TEBERGRzjhUGODJARERkc44RBjiBkIiIqN7YfxjggkNERET1yv7DABccIiIiqlf2Hwa44BAREVG9sv8wcOoUjGiFlIA/KrmAiIiIbMfuw0DSGl+E4AQG/rwCISFAUpLaHRERETkXuw4DRiMw7oN+MEMHADCbgfHjwRECUtWwYcPQpk0beHl5oUWLFnj66afxyy+/WNUcPHgQ9913H7y8vNC6dWssXLiw0n4+/fRTdOjQAV5eXujatSs2bdpk9byI4MUXX0SLFi3g7e2NQYMGITs726rm/PnzGDlyJPz9/REQEID4+HhcvHjR9m+aiJyaXYeB7GzALNYtlpUBOTkqNUQE4IEHHsAnn3yCrKwsfP755zh27Bj+7//+T3m+qKgIgwcPRkhICPbt24dFixZh7ty5WL58uVKza9cuxMbGIj4+Hunp6Rg+fDiGDx+Ow4cPKzULFy7E4sWL8e677yItLQ2+vr6IiYnB1atXlZqRI0fiyJEjSE5OxsaNG7Fjxw6MGzeuYT4IInIeYsdOnRLRokws5xdabjqdZTuRvVi3bp1oNBq5du2aiIi888470rhxYykpKVFqZs2aJe3bt1ceP/744zJ06FCr/URFRcn48eNFRMRsNktwcLAsWrRIeb6wsFA8PT3l448/FhGRjIwMASB79+5Var755hvRaDSSl5dX6/5NJpMAEJPJVId3TUTOxK5HBgwGYHnkIuhQCgDQ6YDERJ5hSPbj/PnzWL16Nfr27Qt3d3cAQGpqKu6//354eHgodTExMcjKysKFCxeUmkGDBlntKyYmBqmpqQCA48ePIz8/36pGr9cjKipKqUlNTUVAQAB69+6t1AwaNAharRZpaWnV9lxSUoKioiKrGxG5NrsOAwAQr/k3chGKlDf3IzcXiI9XuyMiYNasWfD19UWTJk1w8uRJrFu3TnkuPz8fQUFBVvXlj/Pz82usqfh8xddVV9O8eXOr593c3BAYGKjUVGX+/PnQ6/XKrTVX9iRyeXYfBnDmDAzIw4AYT44IUL2ZPXs2NBpNjbfMzEyl/vnnn0d6ejr++9//QqfTYdSoURARFd9B7c2ZMwcmk0m5nSpf4ZOIXJab2g3U6OpVywqEABAcrG4v5NRmzJiBuLi4GmvCw8OVr5s2bYqmTZuiXbt26NixI1q3bo3du3cjOjoawcHBOHPmjNVryx8H3/g5rq6m4vPl21q0aGFV06NHD6WmoKDAah+lpaU4f/688vqqeHp6wtPTs8b3SkSuxb5HBsp/0Xl4AAEBqrZCzq1Zs2bo0KFDjbeKcwAqMpvNACzH4gEgOjoaO3bswPXr15Wa5ORktG/fHo0bN1Zqtm7darWf5ORkREdHAwDCwsIQHBxsVVNUVIS0tDSlJjo6GoWFhdi3b59Ss23bNpjNZkRFRd3pR0JErkTtGYw1SkuznEJgMKjdCZGIiOzevVuWLFki6enpkpubK1u3bpW+fftK27Zt5erVqyJimfUfFBQkTz/9tBw+fFjWrFkjPj4+kpiYqOxn586d4ubmJq+99pocPXpUEhISxN3dXQ4dOqTULFiwQAICAmTdunVy8OBBeeSRRyQsLEyuXLmi1AwZMkR69uwpaWlp8v3330tkZKTExsbW6T3xbAIisu8wsH69JQz06qV2J0QiInLw4EF54IEHJDAwUDw9PSU0NFQmTJggRqPRqu7AgQPSr18/8fT0lFatWsmCBQsq7euTTz6Rdu3aiYeHh3Tu3Fm+/vprq+fNZrO88MILEhQUJJ6envLggw9KVlaWVc25c+ckNjZW/Pz8xN/fX8aMGSPFxcV1ek8MA0SkEbHjWU8rVgBjxwJDhwIbN6rdDZFTKioqgl6vh8lkgr+/v9rtEJEK7HvOQPkEq5tOryIiIiLbYRggIiJycfYdBsoXTmEYICIiqjf2HQbKRwa4xgAREVG9cYwwwJEBIiKiesMwQERE5OLsNwyUlACFhZavGQaIiIjqjf2GgfJRAXd34MYSrkRERGR79h8GgoIAjUbdXoiIiJyYY4QBIiIiqjcMA0RERC7OfsMAFxwiIiJqEPYbBrjgEBERUYOw/zDAkQEiIqJ6xTBARETk4hgGiIiIXJz9hgFOICQiImoQ9hkGKi5FzAmERERE9co+w0BBgeWeSxETERHVO/sMA+XzBZo351LERERE9cy+wwDnCxAREdU7+wwD5ZMHOV+AiIio3tlnGODIABERUYNhGCAiInJxDANEREQujmGAiIjIxdlnGOAEQiIiogZjn2GAIwNEREQNxv7CwLVrwIULlq8ZBoiIiOqd/YWB8qWI3dy4FDEREVEDsL8wUPFqhVr7a4+IiMjZ2N9fW84XICIialAMA0RERC6OYYCIiMjFMQwQERG5OPsLA1xwiIiIqEHZXxjgyAAREVGDYhggIiJycXYXBoyndUjBABilldqtEBERuQS7CgNJ75kRUrgfA5GCkMHtkJSkdkdERETOTyMionYTAGA0AiEhArNZo2zT6YDcXMBgUK8vImdXVFQEvV4Pk8kEf39/tdshIhXYzchAdjasggAAlJUBOTkqNUREROQi7CYMREYCWq31IIVOB0REqNQQERGRi7CbMGAwAMufOQAdSgFYgkBiIg8REBER1Te7CQMAEN9rP3IRipS7ZyI3F4iPV7sjIiIi5+emdgNWzp2DAXkwROYBHBEgIiJqEHY1MoBz5yz3TZqo2wcREZELYRggIiJycQwDRERELo5hgIiIyMUxDBAREbk4hgEiIiIXZz9hQIRhgIiISAX2EwYuXwZKSixfMwwQERE1GPsJA+WjAu7ugJ+fur0QERG5EPsLA02aABpNzbVERERkM/YZBoiIiKjB2E8YOH/ecs8wQERE1KDsJwxwZICIiEgVDANEREQujmGAiIjIxTEMEBERuTiGASIiIhfnpnYDCoYBIrJTZWVluH79utptEFXi7u4OnU53x/thGCAiqoaIID8/H4WFhWq3QlStgIAABAcHQ3MHC/apHgaMRiA7G4j81RMGgGGAHEZJSQmioqJw4MABpKeno0ePHspzBw8exMSJE7F37140a9YMkydPxsyZM61e/+mnn+KFF15Abm4uIiMj8eqrr+Lhhx9WnhcRJCQk4L333kNhYSHuvfdeLFu2DJGRkUrN+fPnMXnyZGzYsAFarRaPPfYY3n77bfhxSW+bKA8CzZs3h4+Pzx39siWyNRHB5cuXUVBQAABo0aLFHe1MNStWiGi1IoCIFqWyAn8WOXNGzZaIam3KlCny0EMPCQBJT09XtptMJgkKCpKRI0fK4cOH5eOPPxZvb29JTExUanbu3Ck6nU4WLlwoGRkZ8o9//EPc3d3l0KFDSs2CBQtEr9fLV199JQcOHJBhw4ZJWFiYXLlyRakZMmSIdO/eXXbv3i3fffedRERESGxsbJ3eh8lkEgBiMplu/8NwQqWlpZKRkSFnz55VuxWiGp09e1YyMjKktLT0tvehWhg4deq3IFB+0+G6nDp+Xa2WiGpt06ZN0qFDBzly5EilMPDOO+9I48aNpaSkRNk2a9Ysad++vfL48ccfl6FDh1rtMyoqSsaPHy8iImazWYKDg2XRokXK84WFheLp6Skff/yxiIhkZGQIANm7d69S880334hGo5G8vLxavxeGgapduXJFMjIy5PLly2q3QlSjy5cvS0ZGhtU/FOpKtbMJsrMBs9l6WxnckJOr+pELohqdOXMGY8eOxQcffAAfH59Kz6empuL++++Hh4eHsi0mJgZZWVm4cOGCUjNo0CCr18XExCA1NRUAcPz4ceTn51vV6PV6REVFKTWpqakICAhA7969lZpBgwZBq9UiLS2t2v5LSkpQVFRkdaPq8dAA2Ttb/IyqFgYiIwHtTd9dh1JERKjTD1FtiAji4uIwYcIEqz/CFeXn5yMoKMhqW/nj/Pz8GmsqPl/xddXVNG/e3Op5Nzc3BAYGKjVVmT9/PvR6vXJr3bp1je+ZiJyfamHAYACWLwfKz4jQoRSJoQtgMKjVEbmy2bNnQ6PR1HjLzMzEkiVLUFxcjDlz5qjd8m2bM2cOTCaTcjt16pTaLRGRylRddCg+HsjNBVJmbUYuQhHfYaea7ZALmzFjBo4ePVrjLTw8HNu2bUNqaio8PT3h5uaGiBtDWb1798bo0aMBAMHBwThz5ozV/ssfBwcH11hT8fmKr6uupnwWcbnS0lKcP39eqamKp6cn/P39rW7kXEQE48aNQ2BgIDQaDfbv319lXVZWFoKDg1FcXFyr/Z49exbNmzeH0Wi0YbdkD1RfgdBgAAY0OwID8nhaIammWbNm6NChQ403Dw8PLF68GAcOHMD+/fuxf/9+bNq0CQCwdu1avPLKKwCA6Oho7Nixw2qRmuTkZLRv3x6NGzdWarZu3WrVQ3JyMqKjowEAYWFhCA4OtqopKipCWlqaUhMdHY3CwkLs27dPqdm2bRvMZjOioqLq4VMiR7F582asWrUKGzduxOnTp9GlSxfExcVh7ty5VnVz5szB5MmT0ahRo1rtt2nTphg1ahQSEhLqoWvbiouLw/Dhw9Vuw2GoHgYA/LbgUGCgun0Q3UKbNm3QpUsX5dauXTsAQNu2bWG4cYzrqaeegoeHB+Lj43HkyBGsXbsWb7/9NqZPn67sZ+rUqdi8eTNef/11ZGZmYu7cufjhhx8wadIkAJYJQdOmTcM///lPrF+/HocOHcKoUaPQsmVL5Rdcx44dMWTIEIwdOxZ79uzBzp07MWnSJDz55JNo2bJlw34wZFeOHTuGFi1aoG/fvggODoabW+WJ2SdPnsTGjRsRFxdXp32PGTMGq1evxvnz523ULdkFm53bcCfGjbOcWzh3rtqdENXJ8ePHK51aKCJy4MAB6devn3h6ekqrVq1kwYIFlV77ySefSLt27cTDw0M6d+4sX3/9tdXzZrNZXnjhBQkKChJPT0958MEHJSsry6rm3LlzEhsbK35+fuLv7y9jxoyR4uLiOr0HnlpYtfJTC61O1zKbRS5eVOdmNteq79GjRwsA5RYSEqJsT0hIUOoWLVokvXv3tnrtmDFjpGvXrnL16lURESkpKZEePXrI008/bVUXFhYmK1asqLGP7777Tvr16ydeXl5iMBhk8uTJcvHiRRERef/998XX11d++uknpf6vf/2rtG/fXi5duiQiIv/5z3+kV69e4ufnJ0FBQRIbGytnblqH5vDhwzJ06FBp1KiR+Pn5Sb9+/SQnJ0cSEhKsPgMAkpKSUqvPzxFV+bNaR/YRBh57zBIGlixRuxMil8MwULUqf8FevGi9OEpD3m78Ib2VwsJCefnll8VgMMjp06eloKBARCqHgWHDhsmECROsXltcXCzh4eEybdo0ERF57rnnJDQ0tNLPxhNPPCGjR4+utoecnBzx9fWVN998U3766SfZuXOn9OzZU+Li4pSaESNGyN133y3Xr1+XjRs3iru7u/zwww/K80lJSbJp0yY5duyYpKamSnR0tDz00EPK80ajUQIDA+XRRx+VvXv3SlZWlvz73/+WzMxMKS4ulscff1yGDBkip0+fltOnT1ut++FsbBEG7OOkfl6XgIjIJvR6PRo1agSdTmc1kXTVqlVWdSdOnKh0eqyfnx8+/PBD9O/fH40aNcJbb72FlJSUSpNMW7ZsifT09Gp7mD9/PkaOHIlp06YBACIjI7F48WL0798fy5Ytg5eXFxITE9GtWzdMmTIFX3zxBebOnYtevXop+/jzn/+sfB0eHo7Fixfj7rvvxsWLF+Hn54elS5dCr9djzZo1cHd3BwDlsB0AeHt7o6SkpMbJtPQbhgEiotry8QEuXlTve9vQlStX4OXlVWl7dHQ0nnvuOcybNw+zZs1Cv379KtV4e3vj8uXL1e77wIEDOHjwIFavXq1sExGYzWYcP34cHTt2ROPGjZGUlISYmBj07dsXs2fPttrHvn37MHfuXBw4cAAXLlyA+cYqdSdPnkSnTp2wf/9+3HfffUoQoDvDMEBEVFsaDeDrq3YXNtG0aVNlRcyKzGYzdu7cCZ1Oh5ycnCpfe/78eTRr1qzafV+8eBHjx4/HlClTKj3Xpk0b5esdO3ZAp9Ph9OnTuHTpknJWw6VLlxATE4OYmBisXr0azZo1w8mTJxETE4Nr164BsAQSsh31zyYQYRggImpgPXv2REZGRqXtixYtQmZmJr799lts3rwZK1eurFRz+PBh9OzZs9p933XXXcjIyEBERESlW/ky3bt27cKrr76KDRs2wM/PTzmTBgAyMzNx7tw5LFiwAPfddx86dOhQaU2Nbt264bvvvrM6hbciDw8PlJWV1eqzIHsIA5cvAyUllq8ZBoiIGkT5tTAq/sFMT0/Hiy++iBUrVuDee+/FG2+8galTp+Lnn39Wai5fvox9+/Zh8ODB1e571qxZ2LVrFyZNmoT9+/cjOzsb69atU/7gFxcX4+mnn8aUKVPw0EMPYfXq1Vi7di0+++wzAJbRAw8PDyxZsgQ///wz1q9fj3nz5ll9j0mTJqGoqAhPPvkkfvjhB2RnZ+ODDz5AVlYWACA0NBQHDx5EVlYWzp49W21ooBtsN5/xNp04YZkp6+5e61NniMh2eDZB1WwxQ1stb775pnJKYXWuX78uLVu2lM2bN4uI5f126tRJxo0bZ1U3bNgw6du3r3J53I8++sjqCpzV2bNnj/zud78TPz8/8fX1lW7duskrr7wiIpVPYRQRef311yUwMFCMRqPyfUJDQ8XT01Oio6Nl/fr1lU7jPXDggAwePFh8fHykUaNGct9998mxY8dERKSgoED5/uCphbekERFRNY2kpwN33QUEBwOnT6vaCpErKioqgl6vh8lk4tLEFVy9ehXHjx9HWFhYlRPtnMHSpUuxfv16bNmypdavueeeezBlyhQ89dRT9dgZ1YUtflbVn0DI+QJERKoYP348CgsLUVxcXKslic+ePYtHH30UsbGxDdAdNSSGASIiF+Xm5oa///3vta5v2rQpZs6cWY8dkVrUn0DIMEBERKQqhgEiIiIXxzBARETk4hgGiIiIXBzDABERkYtjGCAiInJxDANERE5GRDBu3DgEBgZCo9Fg//79VdZlZWUhODgYxcXFtdrv2bNn0bx5cxiNRht2a39yc3Nr/NycEcMAEZGT2bx5M1atWoWNGzfi9OnT6NKlC+Li4jB37lyrujlz5mDy5Mm1WnAIsKwzMGrUKCQkJNRD1/ajdevWyudWn+wpdKgbBkpLgcJCy9cMA0RENnHs2DG0aNECffv2RXBwMNzcKq8vd/LkSWzcuBFxcXF12veYMWOwevVqnD9/3kbd2pYtLkik0+mq/dyclbphoOK1tAMD1euDiKieGY1ASorlvj7FxcVh8uTJOHnyJDQaDUJDQ6us++STT9C9e3e0atVK2fbnP/8Z3bp1Q8mNK8leu3YNPXv2xKhRo5Sazp07o2XLlvjyyy+r7eHcuXOIjY1Fq1at4OPjg65du+Ljjz+2qhkwYAAmTZqESZMmQa/Xo2nTpnjhhRdQ8XI5oaGhmDdvHmJjY+Hr64tWrVph6dKlVvvRaDRYtmwZhg0bBl9fX7zyyisAgGXLlqFt27bw8PBA+/bt8cEHH9T6fd78L/bt27dDo9Fgy5Yt6NmzJ7y9vTFw4EAUFBTgm2++QceOHeHv74+nnnoKly9fVr7P5s2b0a9fPwQEBKBJkyb4/e9/j2PHjinPh4WFAbBcTlqj0WDAgAHKcytWrEDHjh3h5eWFDh064J133qn287YJG1006fYcPWq5YqFer2obRK6MVy2smi2vWrhihYhWa/l1p9VaHteXwsJCefnll8VgMMjp06eloKBARERGjx4tCQkJSt2wYcNkwoQJVq8tLi6W8PBwmTZtmoiIPPfccxIaGlrpZ+OJJ56Q0aNHV9uD0WiURYsWSXp6uhw7dkwWL14sOp1O0tLSlJr+/fuLn5+fTJ06VTIzM+XDDz8UHx8fWb58uVITEhIijRo1kvnz50tWVpayn//+979KDQBp3ry5/Pvf/5Zjx47JiRMn5IsvvhB3d3dZunSpZGVlyeuvvy46nU62bdtWq/d5/PhxqyskpqSkCAC555575Pvvv5cff/xRIiIipH///jJ48GD58ccfZceOHdKkSRNZsGCB0ttnn30mn3/+uWRnZ0t6err84Q9/kK5du0pZWZmIWK7sCED+97//yenTp+XcuXMiIvLhhx9KixYt5PPPP5eff/5ZPv/8cwkMDJRVq1ZV+Xnb4mdV3TDw/feW/zvCw1Vtg8iVMQxUzVZh4NSp34JA+U2ns2yvL7W5hHH37t3l5ZdfrrR9165d4u7uLi+88IK4ubnJd999V6nm2WeflQEDBtSpp6FDh8qMGTOUx/3795eOHTuKucKl62fNmiUdO3ZUHoeEhMiQIUOs9vPEE0/IQw89pDwGoPxRL9e3b18ZO3as1bYRI0bIww8/XKv3WV0Y+N///qfUzJ8/XwAol0wWERk/frzExMRU+xn8+uuvAkAOHTpU5fcp17ZtW/noo4+sts2bN0+io6Or3K8tflbVPUzAyYNE5OSyswGz2XpbWRmQk6NOP+WuXLlS5eVuo6Oj8dxzz2HevHmYMWMG+vXrV6nG29vbajj8ZmVlZZg3bx66du2KwMBA+Pn5YcuWLTh58qRV3T333AONRmP1vbOzs1FWVma17eb+jh49arWtd+/eVo+PHj2Ke++912rbvffea/W62rzPm3Xr1k35OigoCD4+PggPD7faVlBQoDzOzs5GbGwswsPD4e/vrxyyuflzqOjSpUs4duwY4uPj4efnp9z++c9/Wh1isDV1Z0cwDBCRk4uMBLRa60Cg0wEREer1BFjODLhQcd7WDWazGTt37oROp0NONYnl/PnzaNasWbX7XrRoEd5++2289dZb6Nq1K3x9fTFt2jRcu3bNZv1X5OvrW+fX1OZ93szd3V35WqPRWD0u32au8B/6D3/4A0JCQvDee++hZcuWMJvN6NKlS42fw8WLFwEA7733HqKioqye0+l0terzdqg7MlA+G5VhgIiclMEALF9uCQCA5T4x0bJdTT179kRGRkal7YsWLUJmZia+/fZbbN68GStXrqxUc/jwYfTs2bPafe/cuROPPPII/vSnP6F79+4IDw/HTz/9VKkuLS3N6vHu3bsRGRlp9Udv9+7dlWo6duxY43vr2LEjdu7cWamnTp061el93olz584hKysL//jHP/Dggw+iY8eOlcKXh4cHAFiNhAQFBaFly5b4+eefERERYXUrn3BYHzgyQERUz+LjgZgYy6GBiAj1gwAAxMTE4C9/+QvKysqUP77p6el48cUX8dlnn+Hee+/FG2+8galTp6J///7KcPjly5exb98+/Otf/6p235GRkfjss8+wa9cuNG7cGG+88QbOnDlj9ccYsAyXT58+HePHj8ePP/6IJUuW4PXXX7eq2blzJxYuXIjhw4cjOTkZn376Kb7++usa39vzzz+Pxx9/HD179sSgQYOwYcMGfPHFF/jf//5X6/d5pxo3bowmTZpg+fLlaNGiBU6ePInZs2db1TRv3hze3t7YvHkzDAYDvLy8oNfr8dJLL2HKlCnQ6/UYMmQISkpK8MMPP+DChQuYPn26Tfqr5LZnG9hCXp7Ijh2WswqISBWcQFg1W55N0NBqM4Hw+vXr0rJlS9m8ebOIWN5vp06dZNy4cVZ1w4YNk759+0ppaamIiHz00UfSvn37Gvd97tw5eeSRR8TPz0+aN28u//jHP2TUqFHyyCOPKDX9+/eXZ555RiZMmCD+/v7SuHFj+dvf/mY1oTAkJEReeuklGTFihPj4+EhwcLC8/fbbVt8LgHz55ZeVenjnnXckPDxc3N3dpV27dvKf//yn1u+zugmEFy5cUOpXrlwp+pvOhEtISJDu3bsrj5OTk6Vjx47i6ekp3bp1k+3bt1fq97333pPWrVuLVquV/v37K9tXr14tPXr0EA8PD2ncuLHcf//98sUXX1T5edviZ1UjUuGkTiJyOUVFRdDr9TCZTPD391e7Hbtx9epVHD9+HGFhYVVOtHMGS5cuxfr167Fly5Zav+aee+7BlClT8NRTT93R9x4wYAB69OiBt956q9qa0NBQTJs2DdOmTbuj7+XsbPGz6jrLKxERkZXx48ejsLAQxcXFtVqS+OzZs3j00UcRGxvbAN1RQ2IYICJyUW5ubvj73/9e6/qmTZti5syZ9dgRqYVhgIiIGtz27dtvWZObm1vvfZCF+lctJCIiIlUxDBAR1YBzrMne2eJnlGGAiKgK5avL1bTsLpE9KP8ZvXlFxLrgnAEioirodDoEBAQoa837+PhYraNPpDYRweXLl1FQUICAgIA7Wq6YYYCIqBrBwcEAYHXxGSJ7ExAQoPys3i6GASKiamg0GrRo0QLNmzfH9evX1W6HqBJ3d3ebXMCIYYCI6BZ0Ol29XjGOSG2cQEhEROTiGAaIiIhcHMMAERGRi2MYICIicnEMA0RERC5OI1xrk8iliYhyCVsuqkPkmhgGiIiIXBwPExAREbk4hgEiIiIXxzBARETk4hgGiIiIXBzDABERkYtjGCAiInJxDANEREQu7v8DjmbRm/8TpEkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f_composed(x):\n",
    "    return np.exp(-2*x) + 3*np.sin(3*x)\n",
    "\n",
    "plot_f1_and_f2(lambdify(x, dfdx_composed, 'numpy'), np.gradient(f_composed(x_array_2), x_array_2),\n",
    "              label1=\"f'(x) exact\", label2=\"f'(x) approximate\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "826da796",
   "metadata": {},
   "source": [
    "The results are pretty impressive, keeping in mind that it does not matter at all how the function was calculated - only the final values of it!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc60825b",
   "metadata": {},
   "source": [
    "<a name='3.2'></a>\n",
    "### 3.2 - Limitations of Numerical Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8dbf76a0",
   "metadata": {},
   "source": [
    "Obviously, the first downside of the numerical differentiation is that it is not exact. However, the accuracy of it is normally enough for machine learning applications. At this stage there is no need to evaluate errors of the numerical differentiation.\n",
    "\n",
    "Another problem is similar to the one which appeared in the symbolic differentiation: it is inaccurate at the points where there are \"jumps\" of the derivative. Let's compare the exact derivative of the absolute value function and with numerical approximation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "28bb6a5f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def dfdx_abs(x):\n",
    "    if x > 0:\n",
    "        return 1\n",
    "    else:\n",
    "        if x < 0:\n",
    "            return -1\n",
    "        else:\n",
    "            return None\n",
    "\n",
    "plot_f1_and_f2(np.vectorize(dfdx_abs), np.gradient(abs(x_array_2), x_array_2))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "229fdab5",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00ccb653",
   "metadata": {},
   "source": [
    "You can see that the results near the \"jump\" are $0.5$ and $-0.5$, while they should be $1$ and $-1$. These cases can give significant errors in the computations.\n",
    "\n",
    "But the biggest problem with the numerical differentiation is slow speed. It requires function evalutation every time.  In machine learning models there are hundreds of parameters and there are hundreds of derivatives to be calculated, performing full function evaluation every time slows down the computation process. You will see the example of it below."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2caeb33f",
   "metadata": {},
   "source": [
    "<a name='4'></a>\n",
    "## 4 - Automatic Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eba8f444",
   "metadata": {},
   "source": [
    "**Automatic differentiation** (autodiff) method breaks down the function into common functions ($sin$, $cos$, $log$, power functions, etc.), and constructs the computational graph consisting of the basic functions. Then the chain rule is used to compute the derivative at any node of the graph. It is the most commonly used approach in machine learning applications and neural networks, as the computational graph for the function and its derivatives can be built during the construction of the neural network, saving in future computations.\n",
    "\n",
    "The main disadvantage of it is implementational difficulty. However, nowadays there are libraries that are convenient to use, such as [MyGrad](https://mygrad.readthedocs.io/en/latest/index.html), [Autograd](https://autograd.readthedocs.io/en/latest/) and [JAX](https://jax.readthedocs.io/en/latest/). `Autograd` and `JAX` are the most commonly used in the frameworks to build neural networks. `JAX` brings together `Autograd` functionality for optimization problems, and `XLA` (Accelerated Linear Algebra) compiler for parallel computing.\n",
    "\n",
    "The syntax of `Autograd` and `JAX` are slightly different. It would be overwhelming to cover both at this stage. In this notebook you will be performing automatic differentiation using one of them: `JAX`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20071067",
   "metadata": {},
   "source": [
    "<a name='4.1'></a>\n",
    "### 4.1 - Introduction to `JAX`"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f444d827",
   "metadata": {},
   "source": [
    "To begin with, load the required libraries. From `jax` package you need to load just a couple of functions for now (`grad` and `vmap`). Package `jax.numpy` is a wrapped `NumPy`, which pretty much replaces `NumPy` when `JAX` is used. It can be loaded as `np` as if it was an original `NumPy` in most of the cases. However, in this notebook you'll upload it as `jnp` to distinguish them for now."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "85d818ea",
   "metadata": {},
   "outputs": [],
   "source": [
    "from jax import grad, vmap\n",
    "import jax.numpy as jnp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d42ede8e",
   "metadata": {},
   "source": [
    "Create a new `jnp` array and check its type."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "8856647c",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:jax._src.lib.xla_bridge:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Type of NumPy array: <class 'numpy.ndarray'>\n",
      "Type of JAX NumPy array: <class 'jaxlib.xla_extension.DeviceArray'>\n"
     ]
    }
   ],
   "source": [
    "x_array_jnp = jnp.array([1., 2., 3.])\n",
    "\n",
    "print(\"Type of NumPy array:\", type(x_array))\n",
    "print(\"Type of JAX NumPy array:\", type(x_array_jnp))\n",
    "# Please ignore the warning message if it appears."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "730a2dd3",
   "metadata": {},
   "source": [
    "The same array can be created just converting previously defined `x_array = np.array([1, 2, 3])`, although in some cases `JAX` does not operate with integers, thus the values need to be converted to floats. You will see an example of it below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "3008671b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "JAX NumPy array: [1. 2. 3.]\n",
      "Type of JAX NumPy array: <class 'jaxlib.xla_extension.DeviceArray'>\n"
     ]
    }
   ],
   "source": [
    "x_array_jnp = jnp.array(x_array.astype('float32'))\n",
    "print(\"JAX NumPy array:\", x_array_jnp)\n",
    "print(\"Type of JAX NumPy array:\", type(x_array_jnp))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f81ce077",
   "metadata": {},
   "source": [
    "Note, that `jnp` array has a specific type `jaxlib.xla_extension.DeviceArray`. In most of the cases the same operators and functions are applicable to them as in the original `NumPy`, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "742003ec",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2. 4. 6.]\n",
      "3.0\n"
     ]
    }
   ],
   "source": [
    "print(x_array_jnp * 2)\n",
    "print(x_array_jnp[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c7ef8a4",
   "metadata": {},
   "source": [
    "But sometimes working with `jnp` arrays the approach needs to be changed. In the following code, trying to assign a new value to one of the elements, you will get an error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "3fc00cab",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "'<class 'jaxlib.xla_extension.DeviceArray'>' object does not support item assignment. JAX arrays are immutable. Instead of ``x[idx] = y``, use ``x = x.at[idx].set(y)`` or another .at[] method: https://jax.readthedocs.io/en/latest/_autosummary/jax.numpy.ndarray.at.html\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    x_array_jnp[2] = 4.0\n",
    "except TypeError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf9e29fe",
   "metadata": {},
   "source": [
    "To assign a new value to an element in the `jnp` array you need to apply functions `.at[i]`, stating which element to update, and `.set(value)` to set a new value. These functions also operate **out-of-place**, the updated array is returned as a new array and the original array is not modified by the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "ffc53ad2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1. 2. 4.]\n"
     ]
    }
   ],
   "source": [
    "y_array_jnp = x_array_jnp.at[2].set(4.0)\n",
    "print(y_array_jnp)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "05a07ce0",
   "metadata": {},
   "source": [
    "Although, some of the `JAX` functions will work with arrays defined with `np` and `jnp`. In the following code you will get the same result in both lines:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "5b80429d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.        0.6931472 1.0986123]\n",
      "[0.        0.6931472 1.0986123]\n"
     ]
    }
   ],
   "source": [
    "print(jnp.log(x_array))\n",
    "print(jnp.log(x_array_jnp))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89397092",
   "metadata": {},
   "source": [
    "This is probably confusing - which `NumPy` to use then? Usually when `JAX` is used, only `jax.numpy` gets imported as `np`, and used instead of the original one."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20f12b94",
   "metadata": {},
   "source": [
    " <a name='4.2'></a>\n",
    "### 4.2 - Automatic Differentiation with `JAX` "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9cd26792",
   "metadata": {},
   "source": [
    "Time to do automatic differentiation with `JAX`. The following code will calculate the derivative of the previously defined function $f\\left(x\\right) = x^2$ at the point $x = 3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "070e417a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function value at x = 3: 9.0\n",
      "Derivative value at x = 3: 6.0\n"
     ]
    }
   ],
   "source": [
    "print(\"Function value at x = 3:\", f(3.0))\n",
    "print(\"Derivative value at x = 3:\",grad(f)(3.0))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3514bda9",
   "metadata": {},
   "source": [
    "Very easy, right? Keep in mind, please, that this cannot be done using integers. The following code will output an error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "a50295a3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "grad requires real- or complex-valued inputs (input dtype that is a sub-dtype of np.inexact), but got int32. If you want to use Boolean- or integer-valued inputs, use vjp or set allow_int to True.\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    grad(f)(3)\n",
    "except TypeError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "872bbbc6",
   "metadata": {},
   "source": [
    "Try to apply the `grad` function to an array, calculating the derivative for each of its elements: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "caf0e431",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gradient only defined for scalar-output functions. Output had shape: (3,).\n"
     ]
    }
   ],
   "source": [
    "try:\n",
    "    grad(f)(x_array_jnp)\n",
    "except TypeError as err:\n",
    "    print(err)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9452ebc2",
   "metadata": {},
   "source": [
    "There is some broadcasting issue there. You don't need to get into more details of this at this stage, function `vmap` can be used here to solve the problem.\n",
    "\n",
    "*Note*: Broadcasting is covered in the Course 1 of this Specialization \"Linear Algebra\". You can also review it in the documentation [here](https://numpy.org/doc/stable/user/basics.broadcasting.html#:~:text=The%20term%20broadcasting%20describes%20how,that%20they%20have%20compatible%20shapes.)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "f9b28641",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2. 4. 6.]\n"
     ]
    }
   ],
   "source": [
    "dfdx_jax_vmap = vmap(grad(f))(x_array_jnp)\n",
    "print(dfdx_jax_vmap)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "933e382f",
   "metadata": {},
   "source": [
    "Great, now `vmap(grad(f))` can be used to calculate the derivative of function `f` for arrays of larger size and you can plot the output:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "da0a1262",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_f1_and_f2(f, vmap(grad(f)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4162d5e5",
   "metadata": {},
   "source": [
    "In the following code you can comment/uncomment lines to visualize the common derivatives. All of them are found using `JAX` automatic differentiation. The results look pretty good!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "f68b4c0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def g(x):\n",
    "#     return x**3\n",
    "#     return 2*x**3 - 3*x**2 + 5\n",
    "#     return 1/x\n",
    "#     return jnp.exp(x)\n",
    "#     return jnp.log(x)\n",
    "#     return jnp.sin(x)\n",
    "#     return jnp.cos(x)\n",
    "    return jnp.abs(x)\n",
    "#     return jnp.abs(x)+jnp.sin(x)*jnp.cos(x)\n",
    "\n",
    "plot_f1_and_f2(g, vmap(grad(g)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a58ee858",
   "metadata": {},
   "source": [
    "<a name='5'></a>\n",
    "## 5 - Computational Efficiency of Symbolic, Numerical and Automatic Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2211158e",
   "metadata": {},
   "source": [
    "In sections [2.3](#2.3) and [3.2](#3.2) low computational efficiency of symbolic and numerical differentiation was discussed. Now it is time to compare speed of calculations for each of three approaches. Try to find the derivative of the same simple function $f\\left(x\\right) = x^2$ multiple times, evaluating it for an array of a larger size, compare the results and time used:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "36c42dac",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Results\n",
      "Symbolic Differentiation:\n",
      "[-10.       -9.99998  -9.99996 ...   9.99996   9.99998  10.     ]\n",
      "Numerical Differentiation:\n",
      "[-9.99999 -9.99998 -9.99996 ...  9.99996  9.99998  9.99999]\n",
      "Automatic Differentiation:\n",
      "[-10.       -9.99998  -9.99996 ...   9.99996   9.99998  10.     ]\n",
      "\n",
      "\n",
      "Time\n",
      "Symbolic Differentiation:\n",
      "24.546146392822266 ms\n",
      "Numerical Differentiation:\n",
      "272.04132080078125 ms\n",
      "Automatic Differentiation:\n",
      "56.61273002624512 ms\n"
     ]
    }
   ],
   "source": [
    "import timeit, time\n",
    "\n",
    "x_array_large = np.linspace(-5, 5, 1000000)\n",
    "\n",
    "tic_symb = time.time()\n",
    "res_symb = lambdify(x, diff(f(x),x),'numpy')(x_array_large)\n",
    "toc_symb = time.time()\n",
    "time_symb = 1000 * (toc_symb - tic_symb)  # Time in ms.\n",
    "\n",
    "tic_numerical = time.time()\n",
    "res_numerical = np.gradient(f(x_array_large),x_array_large)\n",
    "toc_numerical = time.time()\n",
    "time_numerical = 1000 * (toc_numerical - tic_numerical)\n",
    "\n",
    "tic_jax = time.time()\n",
    "res_jax = vmap(grad(f))(jnp.array(x_array_large.astype('float32')))\n",
    "toc_jax = time.time()\n",
    "time_jax = 1000 * (toc_jax - tic_jax)\n",
    "\n",
    "print(f\"Results\\nSymbolic Differentiation:\\n{res_symb}\\n\" + \n",
    "      f\"Numerical Differentiation:\\n{res_numerical}\\n\" + \n",
    "      f\"Automatic Differentiation:\\n{res_jax}\")\n",
    "\n",
    "print(f\"\\n\\nTime\\nSymbolic Differentiation:\\n{time_symb} ms\\n\" + \n",
    "      f\"Numerical Differentiation:\\n{time_numerical} ms\\n\" + \n",
    "      f\"Automatic Differentiation:\\n{time_jax} ms\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "493e5457",
   "metadata": {},
   "source": [
    "The results are pretty much the same, but the time used is different. Numerical approach is obviously inefficient when differentiation needs to be performed many times, which happens a lot training machine learning models. Symbolic and automatic approach seem to be performing similarly for this simple example. But if the function becomes a little bit more complicated, symbolic computation will experiance significant expression swell and the calculations will slow down.\n",
    "\n",
    "*Note*: Sometimes the execution time results may vary slightly, especially for automatic differentiation. You can run the code above a few time to see different outputs. That does not influence the conclusion that numerical differentiation is slower. `timeit` module can be used more efficiently to evaluate execution time of the codes, but that would unnecessary overcomplicate the codes here.\n",
    "\n",
    "Try to define some polynomial function, which should not be that hard to differentiate, and compare the computation time for its differentiation symbolically and automatically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "13047a93",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Results\n",
      "Symbolic Differentiation:\n",
      "[2.88570423e+24 2.88556400e+24 2.88542377e+24 ... 1.86202587e+22\n",
      " 1.86213384e+22 1.86224181e+22]\n",
      "Automatic Differentiation:\n",
      "[2.8857043e+24 2.8855642e+24 2.8854241e+24 ... 1.8620253e+22 1.8621349e+22\n",
      " 1.8622416e+22]\n",
      "\n",
      "\n",
      "Time\n",
      "Symbolic Differentiation:\n",
      "436.5575313568115 ms\n",
      "Automatic Differentiation:\n",
      "208.57596397399902 ms\n"
     ]
    }
   ],
   "source": [
    "def f_polynomial_simple(x):\n",
    "    return 2*x**3 - 3*x**2 + 5\n",
    "\n",
    "def f_polynomial(x):\n",
    "    for i in range(3):\n",
    "        x = f_polynomial_simple(x)\n",
    "    return x\n",
    "\n",
    "tic_polynomial_symb = time.time()\n",
    "res_polynomial_symb = lambdify(x, diff(f_polynomial(x),x),'numpy')(x_array_large)\n",
    "toc_polynomial_symb = time.time()\n",
    "time_polynomial_symb = 1000 * (toc_polynomial_symb - tic_polynomial_symb)\n",
    "\n",
    "tic_polynomial_jax = time.time()\n",
    "res_polynomial_jax = vmap(grad(f_polynomial))(jnp.array(x_array_large.astype('float32')))\n",
    "toc_polynomial_jax = time.time()\n",
    "time_polynomial_jax = 1000 * (toc_polynomial_jax - tic_polynomial_jax)\n",
    "\n",
    "print(f\"Results\\nSymbolic Differentiation:\\n{res_polynomial_symb}\\n\" + \n",
    "      f\"Automatic Differentiation:\\n{res_polynomial_jax}\")\n",
    "\n",
    "print(f\"\\n\\nTime\\nSymbolic Differentiation:\\n{time_polynomial_symb} ms\\n\" +  \n",
    "      f\"Automatic Differentiation:\\n{time_polynomial_jax} ms\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "231c9da0",
   "metadata": {},
   "source": [
    "Again, the results are similar, but automatic differentiation is times faster. \n",
    "\n",
    "With the increase of function computation graph, the efficiency of automatic differentiation compared to other methods raises, because autodiff method uses chain rule!\n",
    "\n",
    "Congratulations! Now you are equiped with Python tools to perform differentiation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "93dfd229",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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